Chaos in SU(2) Yang-Mills Chern-Simons Matrix Model
CChaos in SU (2) Yang-Mills Chern-Simons Matrix Model
K. Başkan , S. Kürkçüoˇglu
Middle East Technical University, Department of Physics,Dumlupınar Boulevard, 06800, Ankara, Turkey
E-mails: [email protected]@metu.edu.tr
Abstract
We study the effects of addition of Chern-Simons(CS) term in the minimal YangMills(YM) matrix model composed of two × matrices with SU (2) gauge and SO (2) global symmetry. We obtain the Hamiltonian of this system in appropriatecoordinates and demonstrate that its dynamics is sensitive to the values of boththe CS coupling, κ , and the conserved conjugate momentum, p φ , associated to SO (2) symmetry. We examine the behavior of the emerging chaotic dynamics bycomputing the Lyapunov exponents and plotting the Poincaré sections as thesetwo parameters are varied and, in particular, find that the largest Lyapunov ex-ponents evaluated within a range of values of κ are above that is computed at κ = 0 , for κp φ < . a r X i v : . [ h e p - t h ] J a n Introduction
Recently, there has been growing interest in exploring the structure of chaotic dynamicsemerging from the matrix quantum mechanics [1–12], such as the BFSS and the BMN mod-els [13–19] which appear in the DLCQ quantization of M-theory in the flat and the pp-wavebackground, respectively. These models are SU ( N ) gauge theories, describing the dynam-ics of the N -coincident D -branes, in the flat and spherical backgrounds. It is well-knownthat the gravity dual is obtained in the ’t Hooft limit, i.e. at large N and strong Yang-Mills(YM) coupling and describes a phase in which D -branes form a so called black-brane, i.e.a string theoretical black hole [18–20]. While the earlier investigations (and some recentas well ) [21–26] on the quantum mechanical behaviour of these models were performedin the Euclidean time formulation using both analytical perturbative and non-perturbativemethods, in the past few years, there has been increasing interest on accessing the quantumdynamics using real-time formulations [10, 11]. These studies are propelled by a result dueMaldacena-Shenker-Stanford (MSS) [6], which states that under general circumstances, theLyapunov exponent (which is a measure of chaos in both classical and quantum mechanicalsystems) for quantum chaos is bounded, that this bound is controlled by the temperature ofthe system, and given by λ L ≤ πT . It is conjectured that systems which are holographicallydual to the black holes, are expected to be maximally chaotic. This is already demonstratedfor the Sachdev-Ye-Kitaev (SYK) [27] model, and expected to be so for the BFSS model too.Numerical studies reported in [4] found that, for the BFSS model treated at the classicallevel, the largest Lyapunov exponent is given as λ L = 0 . λ (cid:48) t Hooft ) / . This is para-metrically smaller than the MSS bound πT and violates it only temperatures below ≈ . ,while the quantum correction recently evaluated using Gaussian state approximation [11],indicate that the largest Lyapunov exponent vanishes below a non-zero temperature, andhence ensuring that the MSS bound is not violated.It is important to note that not only the BFSS, BMN matrix models, but even their subsec-tors at small values of N appear as non-trivial many-body systems, and we lack a completesolution to these or even for the smallest Yang-Mills (YM) matrix model to date. The lattermay be described as being composed of two × Hermitian matrices with SU (2) gauge and SO (2) global symmetries. It can be obtained by dimensionally reducing the YM theory in to . Classical dynamics of this system was recently investigated in [5] (see alsothe references [28, 29] in this context) and it was shown that, using the SU (2) gauge and SO (2) rotations of the two matrices among themselves and a judicious choice of coordinatesto fully implement the Gauss law constraint leads to a Hamiltonian with two degrees offreedom and their conjugate momenta. In addition, the angular momentum, p φ , associatedto the rigid SO (2) symmetry appears as a conserved quantity via a term proportional to thesquare of p φ and strongly controls the structure of the effective potential and the ensuingdynamics. At p φ = 0 , the model collapses to the usual x y potential, which is already knownto lead to almost completely chaotic dynamics [30–33]. In [5], the response of the system toa range of different values of p φ is investigated and it is found that, at fixed energy, there is value of p φ above which the chaos ceases to exist and the dynamics is essentially describedby quasi-periodic motion. Therefore, the model is conjectured to have two phases, namelya chaotic phase corresponding to a toy model for a black hole, and a phase consisting oftwo D -branes tied with fixed number of open strings stretching between them, with a forcethat depends on the number of excited strings. The latter can be roughly thought of as the"adiabatic invariant" for the quasi-periodic orbits, which appear as the Kolmogorov-Arnold-Moser (KAM) tori (see, for instance, [36]) in the Poincaré sections. For a given value ofenergy these two phases can coexist within a range of values of p φ , while the end of chaoticdynamics is argued to correspond to the end of the black hole phase. Quantum aspects ofthe × matrix model is addressed in [34], where the ground state energy is also estimated.Recently, bootstrapping methods are proposed to study the quantum structure of the lowdimensional matrix models in [35].In order to gain further insight into the matrix model composed of × matrices with SU (2) gauge symmetry, in this paper, we set out to investigate the dynamics in the presenceof the Chern-Simons (CS) term. It is possible to obtain the corresponding action startingfrom the SU (2) Yang-Mills Chern-Simons (YMCS) model in dimensions and reducingit to . In a manner similar to the one followed in [5], while paying attention to thedifferences in the procedure due to the CS term, which is first order in time derivative,we obtain the Hamiltonian of the system. The latter has the same degrees of freedom asthe pure YM model, while the effective potential is governed by not only p φ , but also theCS coupling κ , which enters into the effective potential as κp φ = k π p φ and another term ∝ κ r . Varying κ at different values of p φ , we probe the impact on the chaotic dynamics.Our new findings are two fold. Firstly, we find that at p φ = 0 , values of the largest (andonly) Lyapunov exponent are above that evaluated at k = 0 , approximately within the rangeof values of | k | (cid:46) . This can be attributed to shrinking of the sharp edges of the effectivepotential contours (see figure 1a), but not sustained further for | k | (cid:38) as the harmonic term ∝ κ r starts to dominate and chaotic dynamics gradually declines. The second and moreinteresting effect is due to the κp φ term, which alters the Lyapunov spectrum depending onits sign, in other words, the orientation of p φ matters. For instance, we find the values of thelargest Lyapunov exponent for κp φ < for a range of values of κ at fixed p φ are above thatis evaluated at k = 0 . These results are presented and discussed in detail in section 3, wherethe results obtained from the Lyapunov data are further corroborated via the use of Poincarésections. Section 2 gives the developments leading to the Hamiltonian of the model. Most ofthe details of the calculations in this section are relegated to the appendix for completeness.We summarize our conclusion and briefly comment on future directions of development insection 4. Let us immediately note here that, although the CS coupling is quantized for the non-abelian CS termin dimensions, this is not so after dimensional reduction to ; the CS term is gauge invariant in dimensions. Nevertheless, we use for the CS coupling κ = k π with k not necessarily an integer, as thisconveniently gives the relevant range of κ values with | k | (cid:47) at E = 1 and | p φ | (cid:47) . See sections 2 & 3 forthe full discussion. SU (2) Matrix Model with the Chern-Simons Term
The action may be given as S = S Y M + S CS , (2.1) where S Y M = (cid:90) d t Tr (cid:20)
12 ( D X i ) + 14 (cid:2) X i , X j (cid:3) (cid:21) , (2.2) and S CS = κ (cid:90) dt Tr[ (cid:15) ij ( X i ˙ X j + 2 iA X i X j )] = κ (cid:90) dt Tr[ (cid:15) ij X i ( D X j )] . (2.3) In these expressions X , X are × traceless Hermitian matrices whose entries are functionsof time only. They transform under the adjoint representation of SU (2) : X i → U † X i U asusual. D X i = ∂ X i − i [ A , X i ] are the covariant derivatives, and A is a gauge field whichtransforms accordingly under the local SU (2) gauge group. S is invariant under the local SU (2) gauge symmetry as well as under a global SO (2) ; i.e. the "rigid" rotations of the X i ’samong themselves. In 2.3, κ is the dimensionless CS coupling constant. Note that, due thegauge invariance of S CS term in -dimensions, κ is not level quantized.It is convenient to work in the A = 0 gauge. In the presence of the CS term Gauss lawconstraint then takes the form − (cid:104) X i , ˙ X i (cid:105) + 2 κ(cid:15) ij X i X j = 0 . (2.4) We may express the matrices X i as X i = 1 √ (cid:126) x i · (cid:126)σ = 1 √ x αi σ α , i : 1 , , & α = 1 , , , (2.5) where √ is a normalization factor and σ α are the usual Pauli matrices. For future notationalconvenience, it is also useful to arrange components of X i into column vectors (cid:126) x = x x x , (cid:126) x = x x x (2.6) Substituting (2.5) into the action (2.1) yields the Lagrangian L = 12 ( ˙ (cid:126) x + ˙ (cid:126) x ) + κ ( (cid:126) x · ˙ (cid:126) x − (cid:126) x · ˙ (cid:126) x ) − ( (cid:126) x × (cid:126) x ) , (2.7) while the constraint (2.4) takes the form (cid:126) x × ˙ (cid:126) x + (cid:126) x × ˙ (cid:126) x − κ (cid:126) x × (cid:126) x = 0 . (2.8) The canonical conjugate momenta are easily obtained from the Lagrangian (2.7) as (cid:126) p = ˙ (cid:126) x − κ(cid:126) x ,(cid:126) p = ˙ (cid:126) x + κ(cid:126) x , (2.9)4 hich clearly show that the kinematical and conjugate momenta are no longer the samein the presence of the CS term; a fact which is widely known in the literature (see forinstance, [37]). Using these in 2.8, Gauss law constraint can be expressed as the conditionof the vanishing of the total angular momentum: (cid:126) L = (cid:126) L + (cid:126) L = 0 . (2.10) In order to obtain the corresponding Hamiltonian, we need to observe that the La-grangian involves a term which is first order in time derivatives. Let us note that the genericform of such a Lagrangian can be given as L = 12 g ab ˙ q a ˙ q b + f a ˙ q a − V , (2.11) where g ab is the metric associated to the generalized coordinates q a , f a are functions of thegeneralized coordinates, i.e. f a ≡ f a ( q b ) and V ≡ V ( q a ) is the potential. CorrespondingHamiltonian can be shown to take the form (see the appendix B for details) H = 12 g − ab p a p b + 12 g − ab f a f b − g − ab f a p b + V . (2.12)
Adapting (2.12) to (2.7), in the Cartesian coordinates we obviously have g ab as the Euclideanflat metric, we may write (cid:126) f i = − κε ij (cid:126) x j ( i, j : 1 , ) and observe that V = ( (cid:126) x × (cid:126) x ) . Puttingall these together, we find that the Hamiltonian corresponding to (2.7) takes the form H = 12 ( (cid:126) p + (cid:126) p ) + 12 κ ( (cid:126) x + (cid:126) x ) + κ ( (cid:126) p · (cid:126) x − (cid:126) p · (cid:126) x ) + ( (cid:126) x × (cid:126) x ) , (2.13) with the equations of motion easily evaluated to be ˙ (cid:126) x = p , ˙ (cid:126) p = − κ (cid:126) x + κ(cid:126) p − (cid:126) x × ( (cid:126) x × (cid:126) x ) , ˙ (cid:126) x = p , ˙ (cid:126) p = − κ (cid:126) x − κ(cid:126) p + 2 (cid:126) x × ( (cid:126) x × (cid:126) x ) . (2.14) Using (2.14), the time derivative of (cid:126) L may be expressed as ˙ (cid:126) L = κ (cid:126) x × (cid:126) p − (cid:126) x × ( (cid:126) x × ( (cid:126) x × (cid:126) x )) . (2.15) A similar result holds for ˙ (cid:126) L . Although the second term in 2.15 remains aligned with (cid:126) L asit does in the pure YM matrix model, this is not manifest for the first term. Nevertheless, thesubsequent analysis will show, upon implementing the Gauss law in appropriate coordinates,that the dynamics remain planar.Taking advantage of the local SU (2) ≈ SO (3) and the global SO (2) rotations, we mayintroduce the coordinates ( α, β, γ, r, θ, φ ) . Following [5], we may consider the × matrix M whose columns are the vectors (cid:126) x and (cid:126) x , i.e. M = ( (cid:126) x , (cid:126) x ) and express M as M = 1 √ R ( α, β, γ ) · r r cos θ r sin θ · (cid:32) cos φ sin φ − sin φ cos φ (cid:33) , (2.16)5 here R ( α, β, γ ) is a SO (3) Euler matrix using z − x − z active rotation with the angles ( α, β, γ ) , respectively. Its explicit form is given in the appendix for quick reference. M ≡ ( (cid:126) x T, (0)1 , (cid:126) x T, (0)2 ) with (cid:126) x T, (0)1 := ( r, , and (cid:126) x T, (0)2 := ( r cos θ, r sin θ, may be thought as aconfiguration of the two D -branes oriented coplanarly with a relative angle θ obtainedvia a SU (2) gauge choice. The latter is not preserved in general by SO (2) rotations on (cid:126) x and (cid:126) x , which can be taken to act on the right of M , nor it is preserved by the SO (3) gauge rotations, which acts from the left on M . Thus, taking these facts together, (2.16)is a convenient way to introduce new coordinates for the present dynamical system. Theadvantage of this choice of the coordinates is that, the Gauss law constraint in (2.10) canbe fully imposed on the Hamiltonian expressed in terms of the new variables, as we willdemonstrate in what follows. Let us also remark that, this is essentially the same approachfollowed in [5] except that, we no longer restrict the left SU (2) ≈ SO (3) rotations to an SO (2) subgroup in advance, since, it is not readily seen that ˙ (cid:126) L i ( i = 1 , remain alignedwith (cid:126) L i .Let us note in advance that the components of angular momentum (cid:126) L can be expressedin terms of the conjugate momenta ( p α , p β , p γ ) corresponding to the Euler angles ( α, β, γ ) as [39] (cid:126) L = sin α ( p γ csc β − p α cot β ) + p β cos α cos α csc β ( p α cos β − p γ ) + p β sin αp α . (2.17) which immediately implies that the Gauss law constraint (cid:126) L = 0 is equivalent to p α = p β = p γ = 0 . (2.18) We will make use of (2.18) to fully impose the Gauss law in what follows.The metric in the new coordinates ( r, θ, φ, α, β, γ ) is straightforwardly obtained from theexpression g ij = Tr (cid:16) ∂ i M † ∂ j M (cid:17) . (2.19) We give the components of g ij and its inverse g ij in appendix B and also provide there thedetails of the evaluation of the Hamiltonian in the new coordinates using the generic formin (2.12) together with the inverse metric g ij . Employing these facts and imposing the Gausslaw constraint (2.18) , we find H = 12 p r + 2 r p θ + p φ r cos ( θ ) + κp φ + κ r r sin ( θ ) + (cid:126) r , =: 12 p r + 2 r p θ + V eff . (2.20) Since this Hamiltonian is cyclic in φ , just like as in the pure YM case [5], p φ is a constantof motion and taking advantage of this fact, we have defined the effective potential, V eff As this is not frequently encountered in the literature, we provide a quick derivation in the appendix C. n the second line of (2.20). A number of remarks regarding this Hamiltonian are now inorder. Firstly, we observe that the terms involving the CS coupling κ are new and thereforewe are now in a position to examine the chaotic dynamics emerging from (2.20) as κ andthe angular momentum p φ assume a range of different values. As we have already indicated,there is no level quantization for the CS term in -dimension, however in what followswe use κ = k π with k taking not necessarily an integer, as this is convenient to give therelevant range of κ for the energy and p φ values that we will use in what follows. Also notethe presence of the (cid:126) r term, which is added to V eff , since the dependence of H on θ canstill clearly be regarded as adiabatic along with same line of reasoning provided in [5] forthe pure YM model, with (cid:126) taken as a small parameter. Though, the interesting new fact isthat, for k (cid:54) = 0 , V eff already develops a minimum even at (cid:126) = 0 . This minimum is given by θ = 0 , and the real positive root of the quartic equation κ r + (cid:126) r − p φ = 0 . For (cid:126) = 0 , weobtain r = 2 π / ( p φ k ) / , which yields E > κp φ for κp φ > and simply E > for κp φ < .At κ = 0 , r ∝ p / φ (cid:126) − / , and for a typical value of (cid:126) = 0 . , V eff ≈ . at p φ = 1 [5], whilefor κ (cid:54) = 0 , this minimum shifts upward for κ > and downward for κ < . For instance,we have V eff ≈ . and . at k = 2 and k = − , respectively; this is illustrated in figure1b. In general, the positive shift of the V eff with increasing values of κp φ > reinforces theharmonic term in the potential and they together act to decrease the Lyapunov exponent,while κp φ < gives a window of negative values ( − < k < ), in which we observe a slightincrease in the Lyapunov spectrum as we will be made manifestly in the next section.It is also useful to have the contour plots of the V eff at p φ = 0 , , for various values of k as will refer to them in the next section. These are given in figure 1a, 1c and 1d. Sharpedges in these potential contours near θ ≈ correspond to the flat direction of the pure YMpotential, i.e. to the case of commuting matrices. In the present case, the CS term helps tolift this, as the harmonic term assists to shrink the sharp edges for all values of p φ and alsoact to pull the contours toward closed loops for p φ (cid:54) = 0 . For p φ > , the latter happens fasterfor k > as opposed to k < and vice versa for p φ < . We now explore the chaotic structure of the system governed by (2.20) by studying theLyapunov spectrum and the Poincaré sections.
Setting the energy E = 1 , (cid:126) = 0 . , and letting p φ assume the values , , , which is conve-nient for ease in comparison with the pure YM matrix model results in [5], we obtain theLargest Lyapunov Exponents (LLE), λ L , as the CS coupling takes on a range of values, inwhich the typical behavior of the LLE’s are captured. Our results are obtained after averag-ing over randomly selected initial conditions in each case, which are given in the figures2a, 2b, 2c and we will elaborate on them shortly. Our method for choosing initial conditions is explained in appendix A. a) p φ = 0 and k = ± (b) p φ = 1 and V eff ( r, θ = 0) (c) p φ = 1 and k = − (d) p φ = 1 and k = 2 Figure 1: Contour plots for V eff in ( a ) , ( c ) , ( d ) , V eff ( r, θ = 0) in ( b ) Chaotic structure of the pure YM model is explored in [5] and it is found that the systemis fully chaotic at p φ = 0 and essentially becomes non-chaotic with increasing values of p φ .At the intermediate values < p φ < , for example at p φ = 1 , there are regions in the phasespace, in which quasi-periodic motion is present as signaled by KAM tori appearing in thePoincaré section plots given in [5], while the rest of the phase space is filled with chaoticmotion.In figure 2a, profile of the Lyapunov spectrum of the model for integer and half-integervalues of k in the interval | k | ≤ at p φ = 0 is presented. The plot is essentially symmetricw.r.t. the k = 0 -axis as may be expected from (2.20), which is even under k ↔ − k for p φ = 0 and although LLE values tend to decrease in an almost monotonic manner for | k | > ,they are essentially non-vanishing for | k | ≤ , which makes us conclude that the model ischaotic and behaves similar to the pure YM case within this range of the CS coupling. Therather mild increase in the LLE values observed in this plot in the narrow range | k | < can be explained as follows. As | k | increases, sharp edged regions in the contour plot of theeffective potential V eff in 1a become less pronounced, and consequently, compared to κ = 0 ,it is naturally expected that the system spends less time in these regions where the dynamicsis adiabatic in θ and therefore no appreciable contribution to chaos arises as already argued n [5]. This, therefore, gives a mild increase in the LLE spectrum within the indicated rangeof k values. Nevertheless, for | k | > , harmonic term starts to become significant and thechaotic dynamics is gradually lost. (a) p φ = 0 (b) p φ = 1 (c) p φ = 2 Figure 2: Lyapunov spectra versus k values at p φ = 0 , , . For p φ (cid:54) = 0 , the term κp φ in V eff effects the Lyapunov spectrum asymmetrically dependingon its sign, as it causes a fixed negative or a positive shift on the latter. At p φ = 1 , for instance,which is illustrated in figure 2b, we immediately observe that λ L values with in the rangeof values − < k < are above what is computed at k = 0 . This can be attributed to thedownward shift in V eff due to κp φ < , which clearly also lowers the minimum of V eff aswe have already discussed toward the end of previous section. The increase in λ L can notbe sustained for k < − , since then the harmonic term ∝ k r becomes sufficiently strongeven at short distances to dominate V eff and initiates the decline of the chaotic dynamics.For k > , this term acts to strengthen the harmonic terms and the chaotic motion becomessharply suppressed before k ≈ . At p φ = 2 , we still observe a mild increase in the Lyapunovexponents roughly in the range − < k < , but the maximum value of λ L now appears to e ≈ . , an order of magnitude less than that is found for p φ = 0 and p φ = 1 , and notsignificant enough to conclude that any dense chaotic dynamics remain for p φ ≥ . All of the conclusions of the previous subsection regarding the chaotic dynamics of thismodels are well supported by the Poincaré sections. We have obtained the latter at the θ = 0 intersections of the phase space and projected on to the p θ − p r -plane. Figures 3, 4, 5 and6show Poincaré sections on the first quadrant of the p θ , p r plane.From figure 3, we see that chaotic dynamics appears to fill the phase space at p φ = 0 ,for a large range of values of k , which is approximately | k | (cid:46) , while the periodic motionstarts to compete and take over after this range of k values as can be observed from figure3c. (a) k = ± (b) k = ± (c) k = ± Figure 3: Poincaré sections at p φ = 0 At p φ = 1 and k = 1 , from figure 4, we observe that the phase space is still dominatedby chaos, while a few KAM tori indicating quasi-periodic motion are visible. As k continuesto increase, more KAM tori start to occur, and system swiftly becomes non-chaotic for k (cid:38) and gets dominated by quasi-periodic orbits. However, for k < case illustrated in figure5, the system appears to remain densely chaotic with only a few KAM tori appearing untilaround k ≈ − , while the quasi-periodic motion starts to spread for k (cid:46) − . and start totake over only after k (cid:46) − . Let us also note that, some KAM tori appear to intersect atlarger values of | k | , as readily seen, for instance, in the figures (4d) and (5f). This is due topossible different values of the r coordinate appearing in the evolution of the system startingwith distinct initial conditions being projected to the same point on the p θ , p r plane.For p φ = 2 , we see that there is very little chaos remaining in the phase space regardlessof the value of k and quasi periodic motion dominates the phase space. This can be seen fromthe Poincaré sections in figure 6. There is no chaos for k > , and although some randomlyspread points appear for negative k values, for small | k | , KAM tori quickly dominate thephase space and quasi periodic motion is all that is left. a) k = 1 (b) k = 2 (c) k = 3 (d) k = 4 Figure 4: Poincaré sections at p φ = 1 for k > In this paper, we have studied the chaotic structure of the minimal Yang Mills Chern Si-mons matrix model. Using the gauge and global symmetries, and with a suitable choice ofthe coordinates, the Hamiltonian of the system is obtained in form in which the Gauss lawconstraint is fully imposed. We have studied the chaotic dynamics of the model, and in par-ticular, probed the changes in the Lyapunov exponent as the values of both the CS coupling, κ , and the conserved conjugate momentum, p φ , are varied. We have found that, even for p φ = 0 , there is a range of CS coupling values, approximately given as | k | (cid:46) within whichthe Lyapunov exponent is larger in value compared to that evaluated at k = 0 . We have alsoseen that κp φ term in the effective potential alters the Lyapunov spectrum depending on itssign. We have found that the largest Lyapunov exponents evaluated within a range of valuesof κ are above that is computed at κ = 0 , for κp φ < . These results are discussed in detailin section 3.Let us finally note that bootstrapping methods for matrix quantum mechanics for small(one- and two-) matrix models developed in a recent article by Han et. al. [35] may also besuitable to address the model studied in this paper at the level of quantum mechanics. Wehope to report on any developments along these directions elsewhere. a) k = − (b) k = − (c) k = − (d) k = − . (e) k = − (f) k = − Figure 5: Poincaré sections at p φ = 1 for k < (a) k = 1 (b) k = 2 (c) k = − (d) k = − (e) k = − (f) k = − Figure 6: Poincaré sections at p φ = 2 cknowledgments Part of S.K.’s work was carried out during his sabbatical stay at the physics department ofCCNY of CUNY and he thanks V.P. Nair and D. Karabali for the warm hospitality at CCNY andthe metropolitan area. S.K. thanks A.P.Balachandran for discussions and critical comments.Authors acknowledge the support of TUB˙ITAK under the project number 118F100 and theMETU research project GAP-105-2018-2809.
References [1] Y. Sekino and L. Susskind, JHEP , 065 (2008) [arXiv:0808.2096 [hep-th]].[2] C. Asplund, D. Berenstein and D. Trancanelli, Phys. Rev. Lett. , 171602 (2011)[arXiv:1104.5469 [hep-th]].[3] S. H. Shenker and D. Stanford, JHEP , 067 (2014) [arXiv:1306.0622 [hep-th]].[4] G. Gur-Ari, M. Hanada and S. H. Shenker, JHEP , 091 (2016) [arXiv:1512.00019[hep-th]].[5] D. Berenstein and D. Kawai, Phys. Rev. D , no. 10, 106004 (2017) [arXiv:1608.08972[hep-th]].[6] J. Maldacena, S. H. Shenker and D. Stanford, JHEP , 106 (2016)[arXiv:1503.01409 [hep-th]].[7] S. Aoki, M. Hanada and N. Iizuka, JHEP , 029 (2015) [arXiv:1503.05562 [hep-th]].[8] Y. Asano, D. Kawai and K. Yoshida, JHEP , 191 (2015) [arXiv:1503.04594 [hep-th]].[9] E. Berkowitz, E. Rinaldi, M. Hanada, G. Ishiki, S. Shimasaki and P. Vranas, Phys. Rev.D , no. 9, 094501 (2016) [arXiv:1606.04951 [hep-lat]].[10] P. Buividovich, M. Hanada and A. Schäfer, EPJ Web Conf. , 08006 (2018)[arXiv:1711.05556 [hep-th]].[11] P. V. Buividovich, M. Hanada and A. Schäfer, Phys. Rev. D , no. 4, 046011 (2019)[arXiv:1810.03378 [hep-th]].[12] Ü. H. Co¸skun, S. Kurkcuoglu, G. C. Toga and G. Unal, JHEP , 015 (2018)[arXiv:1806.10524 [hep-th]].[13] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D , 5112 (1997)[hep-th/9610043].[14] B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B , 545 (1988)
15] N. Itzhaki, J. M. Maldacena, J. Sonnenschein and S. Yankielowicz, Phys. Rev. D ,046004 (1998) [arXiv:hep-th/9802042 [hep-th]].[16] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, JHEP , 013 (2002) [hep-th/0202021].[17] K. Dasgupta, M. M. Sheikh-Jabbari and M. Van Raamsdonk, JHEP , 056 (2002)[hep-th/0205185].[18] B. Ydri, arXiv:1708.00734 [hep-th].[19] B. Ydri, Lect. Notes Phys. , pp.1 (2017) [arXiv:1603.00924 [hep-th]].[20] E. Kiritsis, String theory in a nutshell , Princeton University Press, 2007[21] N. Kawahara, J. Nishimura and K. Yoshida, JHEP , 052 (2006) [arXiv:hep-th/0601170 [hep-th]].[22] N. Kawahara, J. Nishimura and S. Takeuchi, JHEP , 097 (2007) [arXiv:0706.3517[hep-th]].[23] R. Delgadillo-Blando, D. O’Connor and B. Ydri, Phys. Rev. Lett. , 201601 (2008)[arXiv:0712.3011 [hep-th]].[24] R. Delgadillo-Blando, D. O’Connor and B. Ydri, JHEP , 049 (2009) [arXiv:0806.0558[hep-th]].[25] Y. Asano, V. G. Filev, S. Kováˇcik and D. O’Connor, JHEP , 152 (2018)[arXiv:1805.05314 [hep-th]].[26] E. Berkowitz, M. Hanada, E. Rinaldi and P. Vranas, JHEP , 124 (2018)[arXiv:1802.02985 [hep-th]].[27] J. Maldacena and D. Stanford, Phys. Rev. D , no.10, 106002 (2016)[arXiv:1604.07818 [hep-th]].[28] D. N. Kabat and P. Pouliot, Phys. Rev. Lett. (1996), 1004-1007[29] V. Kares, Nucl. Phys. B (2004), 53-75[30] S. G. Matinyan, G. K. Savvidy and N. G. Ter-Arutunian Savvidy, Sov. Phys. JETP ,421 (1981) [Zh. Eksp. Teor. Fiz. , 830 (1981)].[31] G. K. Savvidy, Phys. Lett. , 303 (1983).[32] G. K. Savvidy, Nucl. Phys. B , 302 (1984).[33] I. Y. Aref’eva, P. B. Medvedev, O. A. Rytchkov and I. V. Volovich, Chaos Solitons Fractals , 213 (1999) [hep-th/9710032].
34] R. Hübener, Y. Sekino and J. Eisert, JHEP (2015), 166 [arXiv:1403.1392 [quant-ph]].[35] X. Han, S. A. Hartnoll and J. Kruthoff, Phys. Rev. Lett. (2020) no.4, 041601[arXiv:2004.10212 [hep-th]].[36] E. Ott, Chaos in Dynamical Systems , Cambridge University Press, 2 ed., 2002.[37] G. V. Dunne, Aspects of Chern-Simons theory, in
Topological aspects of low dimensionalsystems Les Houches - Ecole d’Ete de Physique Theorique p. 177–263, 1999. [arXiv:hep-th/9902115 [hep-th]].[38] H. Goldstein, C. P. Poole, and J. L Safko,
Classical Mechanics , 3rd Edition, 2001.[39] J. E. Marsden and T. S. Ratiu,
Introduction to Mechanics and Symmetry :A Basic Ex-position of Classical Mechanical Systems , Springer Publishing Company, 2010.
AppendicesA. Initial Conditions
We pick the initial conditions that are used both in the evaluation of the Lyapunov spectrumand the Poincare sections as follows. Both p r and θ are initially taken to be equal to zero.Using the Hamiltonian (2.20) , p θ can be expressed as p θ = 1 √ (cid:115) − k π r − . r − k π p φ r + r − p φ . (A.1) where we have already set E = 1 and (cid:126) = 0 . . We determine the intervals of r values, whichmake the argument of the square root in (A.1) positive and restrict to the one in which r > . Initial value of r is chosen randomly from this interval and the initial value of p θ isthen determined from (A.1) .We run a Matlab code evaluating the LLE for randomly selected initial conditionsaccording to this procedure at each value of k and take their average to obtain the each datapoint. The error bars are obtained by computing the mean square variances. B. Generic form of the Hamiltonian
For a system with generalized coordinates q i and velocities ˙ q i , Lagrangian involving firstorder time derivatives have the generic form L ( q i , ˙ q i , t ) = 12 g ij ˙ q i ˙ q j + f i ˙ q i − V , (B.1) where g ij is the metric, f i is some function of the generalized coordinates i.e. f i ≡ f i ( q j ) and V is a potential V ≡ V ( q i ) . Canonical momenta are evaluated as p i = ∂L∂ ˙ q i = g ij ˙ q j + f i . (B.2)15 n terms of p i , ˙ q i can be solved using the inverse metric in the form ˙ q i = g − ij ( p j − f j ) , = g ij ( p j − f j ) . (B.3) The Hamiltonian then takes the form H = p i ˙ q i − L = p i g − ij ( p j − f j ) − g ij g − ik ( p k − f k ) g − jl ( p l − f l ) − f i g − ij ( p j − f j ) + V = p i g − ij ( p j − f j ) − g − jl ( p j − f j )( p l − f l ) − f i g − ij ( p j − f j ) + V = p i g − ij ( p j − f j ) − g − ij ( p i − f i )( p j − f j ) − f i g − ij ( p j − f j ) + V = g − ij ( p i p j − p i f j − p i p j + p i f j − f i f j − p i f j + f i f j ) + V = 12 g − ij p i p j + 12 g − ij f i f j − g − ij p i f j + V , (B.4) given in (2.12) . C. Derivation of the Hamiltonian in the New Coordinates
C.1. Metric
A general SO (3) element in the Euler’s parametrization with z − x − z active rotation withthe angles α, β, γ respectively is given by [38] R ( α, β, γ ) = c ( α ) c ( γ ) − s ( α ) c ( β ) s ( γ ) − s ( α ) c ( β ) c ( γ ) − c ( α ) s ( γ ) s ( α ) s ( β ) c ( γ ) s ( α ) + c ( α ) c ( β ) s ( γ ) c ( α ) c ( β ) c ( γ ) − s ( α ) s ( γ ) − c ( α ) s ( β ) s ( β ) s ( γ ) s ( β ) c ( γ ) c ( β ) , (C.1) where s and c stand for sine and cosine, respectively. This can be facilitated to obtain thematrix M in 2.16.The metric in the new coordinates ( r, θ, φ, α, β, γ ) is evaluated using g ij = Tr (cid:16) ∂ i M † ∂ j M (cid:17) and it yields g ij = r r sin( θ ) r cos( β ) 0 r r sin( θ ) r r cos( β ) sin( θ ) 0 r sin( θ )0 r cos( β ) g g g r cos( β )0 0 0 g g r r sin( θ ) r cos( β ) 0 r , (C.2)16 here g = r cos( β ) sin( θ ) ,g = − r cos(2 β ) cos(2( γ + θ )) − r cos(2( β − γ )) − r cos(2( β + γ )) + 14 r cos(2 β )+ 14 r cos( θ ) cos(2 γ + θ ) + 3 r ,g = − r sin( β ) cos( θ ) sin(2 γ + θ ) ,g = − r sin( β ) cos( θ ) sin(2 γ + θ ) ,g = − r cos(2( γ + θ )) − r cos(2 γ ) + r . (C.3) The inverse metric g − ij is given as g ij = r − r sec ( θ ) r − sec( θ ) tan( θ ) r g g g g g g − r − sec( θ ) tan( θ ) r g g g , (C.4) where g = − (cos(2 γ ) + cos(2( γ + θ )) −
2) csc ( β ) csc ( θ ) r ,g = 2 cot( θ ) csc( β ) csc( θ ) sin(2 γ + θ ) r ,g = (cos(2 γ ) + cos(2( γ + θ )) −
2) cot( β ) csc( β ) csc ( θ ) r ,g = 2 cot( θ ) csc( β ) csc( θ ) sin(2 γ + θ ) r ,g = (cos(2 γ ) + cos(2( γ + θ )) + 2) csc ( θ ) r ,g = − β ) cot( θ ) csc( θ ) sin(2 γ + θ ) r ,g = (cos(2 γ ) + cos(2( γ + θ )) −
2) cot( β ) csc( β ) csc ( θ ) r ,g = − β ) cot( θ ) csc( θ ) sin(2 γ + θ ) r ,g = − (cos(2 γ ) + cos(2( γ + θ )) −
2) cot ( β ) csc ( θ ) + sec ( θ ) + 1 r . (C.5)17 .2. Hamiltonian in the new coordinates Corresponding to the generalized coordinates ( r, θ, φ, α, β, γ ) , we label the associated con-jugate momenta as ( p r , p θ , p φ , p α , p β , p γ ) . Using the inverse metric in C.4, we have the firstterm in the generic form of the Hamiltonian (2.12) given as g − ij p i p j = − csc ( β ) cos(2 γ ) csc ( θ ) p α r − csc ( β ) csc ( θ ) p α cos(2( γ + θ ))2 r − β ) csc( β ) csc ( θ ) p α p γ r + cot( β ) csc( β ) cos(2 γ ) csc ( θ ) p α p γ r + cot( β ) csc( β ) csc ( θ ) p α p γ cos(2( γ + θ )) r + 2 csc( β ) cot( θ ) csc( θ ) p α p β sin(2 γ + θ ) r + csc ( β ) csc ( θ ) p α r + cot ( β ) csc ( θ ) p γ r − cot ( β ) cot( θ ) csc( θ ) p γ cos(2 γ + θ ) r + cot( θ ) csc( θ ) p β cos(2 γ + θ ) r − β ) cot( θ ) csc( θ ) p β p γ sin(2 γ + θ ) r + csc ( θ ) p β r − tan( θ ) sec( θ ) p γ p φ r − p γ p θ r + sec ( θ ) p γ r + p γ r + sec ( θ ) p φ r + 2 p θ r + p r . (C.6) In order to proceed, we need to evaluate the form of (cid:126) f i = − κε ij (cid:126) x j ( i, j : 1 , ) in the newcoordinates. The function f i = f i ( q j ) and ˙ q i in the Lagrangian (2.7) appear as f i ˙ q i ≡ κ(cid:126) x · ˙ (cid:126) x − κ(cid:126) x · ˙ (cid:126) x . (C.7) Since the i th column of the matrix M in 2.16 correspond to the components of (cid:126) x i , and sodoes the correspondence goes with their time derivatives, right hand side of (C.7) can bewritten by taking the inner products of the column vectors of M and ˙ M and this yields f i ˙ q i = − r κ (cid:16) φ + sin( θ )(2 ˙ α cos( β ) + 2 ˙ γ + ˙ θ ) (cid:17) . (C.8)18 ince f i ˙ q i = f ˙ r + f ˙ θ + f ˙ φ + f ˙ α + f ˙ β + f ˙ γ in the new coordinates, the coefficients f i ( r, θ, φ, α, β, γ ) , ( i : 1 , ..., are now easily read out from (C.8) to be f = 0 ,f = − r κ sin( θ ) ,f = − r κ ,f = − r κ sin( θ ) cos( β ) ,f = 0 ,f = − r κ sin( θ ) . (C.9) With f i given (C.9) , we can evaluate the second and the third term in (2.12) . We find g − ij f i f j = 12 r κ , (C.10) and g − ij p i f j = − p φ κ , (C.11) The last term takes the form
12 ( (cid:126) x × (cid:126) x ) = 14 r sin ( θ ) , (C.12) which is the same as that would be obtained had we used the matrix M , since the square ofthe cross product of the column vectors of M is a scalar and does not get affected by gaugerotations.Putting (C.6) , (C.10) , (C.11) , (C.12) together and imposing the Gauss law constraint (cid:126) L = 0 via (2.18) as p α = p β = p γ = 0 . (C.13) we finally obtain the Hamiltonian given in (2.20) . D. Angular Momentum Vector in terms of Euler Angles and Conjugate Mo-menta
Angular velocities can be expressed in terms of Euler angles and their time derivatives as[38, 39] w = ˙ γ sin( α ) sin( β ) + ˙ β cos( α ) ,w = ˙ β sin( α ) − ˙ γ cos( α ) sin( β ) ,w = ˙ α + ˙ γ cos( β ) . (D.1) In terms of Euler angles and their time derivatives rotational kinetic energy takes the orm T = 12 ( I w + I w + I w )= I ( ˙ γ sin ( α ) sin ( β ) + 2 ˙ β ˙ γ sin( α ) cos( α ) sin( β ) + ˙ β cos ( α ))+ I ( ˙ γ cos ( α ) sin ( β ) − β ˙ γ sin( α ) cos( α ) sin( β ) + ˙ β sin ( α ))+ I (2 ˙ α ˙ γ cos( β ) + ˙ α + ˙ γ cos ( β )) , (D.2) where I i are the moment of inertia with respect to the principal axes associated to z − x − z active rotation with the angles α, β, γ .Momentum conjugate to the Euler angles α, β, γ are p α = ∂T∂ ˙ α , p β = ∂T∂ ˙ β , p γ = ∂T∂ ˙ γ , (D.3) and we have p α p β p γ = I I cos( β )0 A A I cos( β ) A A ˙ α ˙ β ˙ γ (D.4) where the the remaining components of the matrix A are given as A = I cos ( α ) + I sin ( α ) ,A = I cos( α ) sin( α ) sin( β ) − I cos( α ) sin( α ) sin( β ) ,A = I cos( α ) sin( α ) sin( β ) − I cos( α ) sin( α ) sin( β ) ,A = I cos ( β ) + I cos ( α ) sin ( β ) + I sin ( α ) sin ( β ) . (D.5) Writing (D.4) as P = A Θ in short, the column matrix ˙Θ = ( ˙ α, ˙ β, ˙ γ ) T can be obtained fromthe equation ˙Θ = A − P . Substituting ˙ α, ˙ β, ˙ γ obtained from this equation into (D.1) yields w , w , w in terms of p α , p β , p γ . Finally, we obtain the components of angular momentum (cid:126) L using L i = ∂T∂w i = I i w i (no sum over i ) as [39] (cid:126) L = sin( α )( p γ csc( β ) − p α cot( β )) + p β cos( α )cos( α ) csc( β )( p α cos( β ) − p γ ) + p β sin( α ) p α . (D.6) Therefore, the Gauss Law constraint (cid:126) L = 0 is equivalent to (2.18) ..